Factorisation — Class 8 Mathematics

1. What Is Factorisation?

Factorisation is the REVERSE of multiplication — writing an expression as a PRODUCT of its FACTORS. If (x+2)(x+3) = x²+5x+6, then factorising x²+5x+6 gives (x+2)(x+3). 'Expanding multiplies factors to get an expression. Factorising breaks an expression into factors. They are INVERSE operations.'

2. Method 1 — Common Factor

Find the HCF of all terms. Take it OUTSIDE the bracket. ALWAYS check for a common factor FIRST.

Example: 12x²y + 18xy² = 6xy(2x + 3y). HCF of 12x²y and 18xy²: coefficients: HCF(12,18)=6. Variables: x (lowest power = x¹) and y (lowest power = y¹). 'Taking out the common factor is the most BASIC method — and the most FREQUENTLY forgotten. Always do this FIRST before trying other methods.'

Worked Examples

• 7a² − 14a = 7a(a − 2).
• 3x³ − 6x² + 9x = 3x(x² − 2x + 3).
• pq + pr − qs − rs = p(q+r) − s(q+r) = (q+r)(p−s). 'This is factorisation by GROUPING — group terms with common factors.'

3. Method 2 — Factorisation Using Identities

Identity 1: a² + 2ab + b² = (a+b)²

Factorise x² + 8x + 16. Compare with a²+2ab+b²: a=x, b=4 (since 2×x×4=8x, and 4²=16). = (x+4)².

Identity 2: a² − 2ab + b² = (a−b)²

Factorise x² − 10x + 25. a=x, b=5 (since 2×x×5=10x, and 5²=25). = (x−5)².

Identity 3: a² − b² = (a+b)(a−b) — Difference of Squares

Factorise x² − 49 = (x+7)(x−7). 4y² − 25z² = (2y)²−(5z)² = (2y+5z)(2y−5z). x⁴ − 16 = (x²)²−4² = (x²+4)(x²−4) = (x²+4)(x+2)(x−2). 'Apply difference of squares TWICE if needed — factorise each resulting difference of squares further.'

4. Method 3 — Splitting the Middle Term (Quadratic Trinomial)

For x² + bx + c: find two numbers p and q such that p + q = b and p × q = c. Then: x² + bx + c = x² + px + qx + c = x(x+p) + q(x...). Actually: = (x+p)(x+q).

How to find p and q: List ALL factor pairs of the constant term c. Check which pair sums to b. Take that pair as p and q.

Worked Examples

Example 1: Factorise x² + 7x + 12. Find p,q: p+q=7, p×q=12. Factor pairs of 12: (1,12)→sum=13. (2,6)→8. (3,4)→7 ✓. So p=3, q=4. x²+7x+12 = (x+3)(x+4).

Example 2: Factorise x² − 5x + 6. Need p+q=−5, p×q=6. Factor pairs of 6: (−2,−3)→sum=−5 ✓. x²−5x+6 = (x−2)(x−3).

Example 3: Factorise x² − 2x − 15. Need p+q=−2, p×q=−15. Factor pairs of −15: (−5,3)→sum=−2 ✓. x²−2x−15 = (x−5)(x+3).

Example 4: Factorise 6x² + 11x + 3. For coefficient of x² ≠ 1: multiply a×c = 6×3 = 18. Find p,q: p+q=11, p×q=18. (2,9)→sum=11 ✓. Split: 6x²+2x+9x+3 = 2x(3x+1)+3(3x+1) = (3x+1)(2x+3).

5. Division of Algebraic Expressions

Dividing a Monomial by a Monomial

12x³y² ÷ 4xy = (12/4)(x³/x)(y²/y) = 3x²y.

Dividing a Polynomial by a Monomial

(15x³−10x²+5x) ÷ 5x = 15x³/5x − 10x²/5x + 5x/5x = 3x² − 2x + 1.

Dividing a Polynomial by a Polynomial

(x²+7x+12) ÷ (x+3). Since x²+7x+12 factorises as (x+3)(x+4), division by (x+3) gives (x+4). 'Factorise the dividend FIRST. Then cancel common factors with the divisor. This is much easier than long division.'

6. Error Spotting in Factorisation

7. Common Mistakes

1. Not checking for a common factor FIRST: 6x²+12x = 6x(x+2). If you jump to splitting the middle term, you don't need to — common factor is simpler.
2. Sign error in splitting: x²+5x+6 → p and q must both be POSITIVE (sum positive, product positive).
3. Difference of squares ≠ (a−b)²: x²−25 = (x+5)(x−5), NOT (x−5)².

8. AP Exam Focus

Quick Self-Test

1. Factorise: 15pq − 25p²q. (Answer: 5pq(3−5p).)
2. Factorise: x² − 64. (Answer: (x+8)(x−8).)
3. Factorise: x² + 9x + 20. (Answer: (x+4)(x+5).)
4. Factorise: x² − 7x + 12. (Answer: (x−3)(x−4).)
5. Divide: (x²+5x+6) ÷ (x+2). (Answer: x+3.)

Advanced Factorisation Worked Examples

Example — Factorise 49x²−121y²: This is a²−b² with a=7x, b=11y. = (7x+11y)(7x−11y).

Example — Factorise x⁴−81: Treat x⁴ as (x²)² and 81 as 9²: (x²+9)(x²−9). x²−9 can be factorised further: (x+3)(x−3). Final: (x²+9)(x+3)(x−3). 'x²+9 CANNOT be factorised over real numbers (sum of squares, not difference).'

Example — Factorise 2x²+7x+3: Multiply a×c = 2×3=6. Find p,q: p+q=7, p×q=6. (1,6)→sum=7 ✓. Split: 2x²+x+6x+3 = x(2x+1)+3(2x+1) = (2x+1)(x+3).

Example — Factorise 3x²−x−4: a×c = 3×(−4)=−12. p+q=−1, p×q=−12. (3,−4)→sum=−1 ✓. Split: 3x²+3x−4x−4 = 3x(x+1)−4(x+1) = (x+1)(3x−4).

Factorisation Strategy Flowchart

1. Is there a COMMON FACTOR? → Take it out.
2. Is it a PERFECT SQUARE (a²±2ab+b²)? → Use Identity 1 or 2.
3. Is it a DIFFERENCE OF SQUARES (a²−b²)? → Use Identity 3.
4. Is it x²+bx+c (coefficient of x² is 1)? → Split middle term (sum=b, product=c).
5. Is it ax²+bx+c (a≠1)? → Multiply a×c first. Then split.
6. Is it a SUM/DIFFERENCE OF CUBES? → Use a³±b³ identity (AP Class 9 topic).

Error-Spotting Exercise

Find the error: A student factorised x²+4x+4 as (x+4)(x+1). Check: (x+4)(x+1) = x²+5x+4, NOT x²+4x+4. Correct: x²+4x+4 = (x+2)². 'The student found factors that multiply to 4 (correct) but sum to 5 (wrong — should sum to 4).'

Find the error: 4x²−25 = (2x−5)². Check: (2x−5)² = 4x²−20x+25. NOT the same. Correct: 4x²−25 = (2x+5)(2x−5). 'Difference of squares is NOT a perfect square. (a−b)² ≠ a²−b².'

Division — Long Division of Polynomials

Divide (x³−6x²+11x−6) by (x−2). Step 1: x³/x = x². Multiply: x²(x−2)=x³−2x². Subtract: −6x²−(−2x²) = −4x². Bring down 11x: −4x²+11x. Step 2: −4x²/x = −4x. Multiply: −4x(x−2)=−4x²+8x. Subtract: 11x−8x=3x. Bring down −6: 3x−6. Step 3: 3x/x=3. Multiply: 3(x−2)=3x−6. Subtract: 0. Quotient = x²−4x+3. Factorise further: (x−1)(x−3). So (x³−6x²+11x−6)÷(x−2) = (x−1)(x−3).

Key Exam Tips

• Always VERIFY by expanding: multiply your factors back. They should equal the original expression.
• Write 'Factorisation:' at the start of your answer.
• If the expression cannot be factorised further, write 'Cannot be factorised further over real numbers.'
• For division: write both the factorised form of dividend AND divisor. Cancel common factors.